When analyzing the PDP of hyperparameters, we are usually interested in how their values \(\lambda _S\)  impact model performance uniformly across the hyperparameter space. In line with prior work [1]}, we therefore assume \({P}\)  to be the uniform distribution over \(\Lambda _C\) .
Computing \(c_S(\lambda _S)\)  exactly is usually not possible because \(c\)  is unknown and expensive to evaluate in the context of HPO. Thus, the posterior mean \(\hat{m}\)  of the probabilistic surrogate model \(\hat{c}(\lambda )\)  is commonly used as a proxy for \(c\) . Furthermore, the integral may not be analytically tractable for arbitrary surrogate models \(\hat{c}\) . Hence, the integral is approximated by Monte Carlo integration, i.e.,
\(\hat{c}_S\left(\lambda _S\right) &=& \frac{1}{n} \sum \nolimits _{i = 1}^n \hat{m}\left(\lambda _S, \lambda _C^{(i)}\right)\) 
